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Real-Time Fair Resource Allocation in Distributed
Software Defined Networks
Zaid Allybokus, Konstantin Avrachenkov, Jérémie Leguay, Lorenzo Maggi
To cite this version:
Zaid Allybokus, Konstantin Avrachenkov, Jérémie Leguay, Lorenzo Maggi. Real-Time Fair Resource
Allocation in Distributed Software Defined Networks. ITC 29 - 2017 29th International Teletraffic
Congress , Sep 2017, Genoa, Italy. �hal-01652533�
Real-Time Fair Resource Allocation in
Distributed Software Defined Networks
Zaid Allybokus
∗†, Konstantin Avrachenkov
†, Jérémie Leguay
∗, Lorenzo Maggi
∗∗Huawei Technologies, France Research Center
{zaid.allybokus, jeremie.leguay, lorenzo.maggi}@huawei.com
†INRIA, Sophia Antipolis
avratch@inria.fr
Abstract—The performance of computer networks relies on how bandwidth is shared among different flows. Fair resource allocation is a challenging problem particularly when the flows evolve over time. To address this issue, bandwidth sharing techniques that quickly react to the traffic fluctuations are of interest, especially in large scale settings with hundreds of nodes and thousands of flows. In this context, we propose a distributed algorithm based on the Alternating Direction Method of Multipliers (ADMM) that tackles the fair resource allocation problem in a distributed SDN control architecture. Our ADMM-based algorithm continuously generates a sequence of resource allocation solutions converging to the fair allocation while al-ways remaining feasible, a property that standard primal-dual decomposition methods often lack. Thanks to the distribution of all computer intensive operations, we demonstrate that we can handle large instances in real-time.
Index Terms—Software-Defined Networking; Fair Resource Allocation; Alternating Direction Method of Multipliers; α-Fairness; Distributed Algorithms; Distributed SDN Control Plane.
I. INTRODUCTION
Software Defined Networking (SDN) technologies are rad-ically transforming network architectures by offloading the control plane (e.g., routing, resource allocation) to powerful remote platforms that gather and keep a local or global view of the network status in real-time and push consistent config-uration updates to the network equipment. The computation power of SDN controllers fosters the development of a new generation of control plane architecture that uses compute-intensive operations. Initial design of SDN architectures [24] had envisioned the use of one central controller. However, for obvious scalability and resiliency reasons, the industry has quickly realized that the SDN control plane needs to be partially distributed in large network scenarios [11]. Hence, although logically centralized, in practice the control plane may consist of multiple controllers each in charge of a SDN domain of the network and operating together, in a flat [22] or hierarchical [7] architecture.
In this paper, we study the problem of computing a globally fair (in the sense of α-fairness defined by Mo and Walrand in [16], see Section III) resource allocation in a distributed SDN scenario, where the control plane is distributed over several domain controllers. In this context, flows transiting in the network typically correspond to traffic aggregates of a customer or a class of customers. We consider the traffic engineering use case where the size of flows evolves over time and the bandwidth reserved to each of them has to be quickly adjusted towards the novel fair solution.
In distributed SDN architectures, each controller has full information about its own domain. Moreover, it can commu-nicate with adjacent peer controllers and/or with a central, upper-layer controller entity. However, exchanges between controllers are expensive in terms of communication delay and overhead [20]. This technological limitation translates directly into an algorithmic constraint: distributed algorithms for SDN have a limited budget in terms number of iterations to reach convergence.
A second crucial property for any distributed algorithm for SDN is responsiveness. In fact, the network state may be affected by abrupt changes, e.g., flow size variation, flow ar-rival/departure, link/node congestion. In this case, convergence for the previous network state may not even be attained when a change occurs in the system. For this reason, it is often preferable to have a quick access to a good quality solution rather than a provably asymptotically optimal solution with poor convergence rate. Hence, it is crucial that the resource allocation computed by a distributed algorithm is feasible, thus implementable, at any iteration.
To recap, we identify two main requirements for a dis-tributed algorithm for fair resource allocation, namely i) con-verging to a “good” fair solution in a small number of iterations and ii) producing feasible solutions at all iterations. We claim that none of the current methods that allocate resources in an SDN scenario is able to achieve the two aforementioned goals at the same time. Local mechanisms such as Auto-Bandwidth [18] have been proposed to greedily and distributedly adjust the allocated bandwidth to support time-varying IP traffic in Multi Protocol Label Switching (MPLS) networks. Auto-Bandwidth successfully tackles goal ii) but not i), as it neither ensures fairness nor optimizes resources globally. Also, classic primal-dual algorithms have been proposed to solve the α-fair resource allocation problem in distributed SDN scenarios, as in [15]. However, primal-dual algorithms are known to fail at providing feasible solutions at any iteration step, thus they fail at achieving goal ii).
Recently in the optimization research community, the Al-ternating Direction Method of Multipliers (ADMM) [3] has captured the attention for its separability and fast conver-gence properties. We claim that ADMM offers new and yet unexploited possibilities to tackle concurrently the goals i), ii). Indeed, in this paper we show how ADMM serves our purposes, by allowing all controllers to handle their own domains simultaneously, while still converging to a global optimum in the fashion of a general distributed consensus
problem.
Main contributions: We develop an ADMM-based method (FD-ADMM, Algorithm 2) for the α-fair resource allocation problem over a distributed SDN control plane. It iteratively produces resource allocations that converge to the α-fair optimal allocation. Heavy computations, requiring projections on polytopes, can be massively parallelized on a link-by-link basis (Algorithm 2, line 7) in each domain. This yields a convergence rate that does not depend on the partitioning of the network into domains.
We show that our FD-ADMM algorithm can function in real-time, as i) close-to-optimal solutions are available since the very first iterations and ii) feasible allocations are available at all iterations (Proposition 1), a property that standard primal-dual decomposition methods generally lack. This per-mits to adjust within up to a few milliseconds the bandwidth of flows that evolve quickly and need immediate response. Moreover, we show how to achieve a near-optimal conver-gence rate by providing an explicit and adaptive tuning for the FD-ADMM penalty parameter (Scheme 1).
Finally, we benchmark FD-ADMM with the State-of-the-Art (SoA) approach for computing α-fair resource allocations in distributed scenarios in [15], which is based on the Lagrangian dual splitting method. We show that our algorithm outperforms the SoA in terms of convergence rate, feasibility preservation and hence, overall, real-time responsiveness.
The remainder of this paper is organized as follows. Sec-tion II surveys the related work around the fair resource allocation problem. Section III formulates the α-fair resource allocation problem and presents the basic centralized appli-cation of ADMM. Section IV introduces FD-ADMM, our distributed ADMM-based algorithm that benefits from the distribution of SDN controllers over multiple domains. FD-ADMM is based on a reformulation of the problem in the fashion of a general distributed consensus problem. Section V discusses on a near-optimal tuning of the penalty parameter in FD-ADMM. Section VI provides simulations that validate our approach and finally, Section VII concludes the paper.
For the sake of conciseness, we deferred some of our proofs to the technical report in [1].
II. RELATED WORK
The concept of fair resource allocation has been a central topic in networking. Particularly, max-min fairness1 has been
the classic resource sharing principle [2] and has been studied extensively. The concept of proportional fairness and its weighted variants were introduced in [10]. Later, a spectrum of fairness metrics including the two former ones was introduced by Mo and Walrand in [16] as the family of α-fair utility functions.
Some early notable works on max-min fairness include [4], where the authors propose an asynchronous distributed algorithm that communicates explicitly with the sources and pays some overhead in exchange for more robustness and
1A resource allocation strategy is said to be max-min fair if no route can
increase its allocation while remaining feasible without penalizing another route that has a smaller or equal allocation
faster convergence. Later in [21], a distributed algorithm is defined for the weighted variant of max-min fair resource allocation problem in MPLS networks, based on the well-known property that an allocation is max-min fair if and only if each Label-Switched Path (LSP) either admits a bottleneck link amongst its used links or meets its maximal bandwidth requirement (see Definition 4 there of a bottleneck link). The problem of Network Utility Maximization (NUM) was also addressed with standard decomposition methods that could give efficient and very simple algorithms based on gradient ascent schemes performing their update rules in parallel. In this context, Voice [25], then McCormick et al. [15], tackle the α-fair resource allocation problem with a gradient descent applied to the dual of the problem.
In these works, no mention is made on the potential (in fact, systematic) feasibility violation of the sequences generated by those algorithms, which is a crucial matter in distributed SDN settings. Regarding this topic, the authors of [12] employ damping techniques to avoid transient infeasibility while reaching the max-min fair point, but cannot guarantee feasibility at all times, especially in dynamic settings. Also motivated by this, more recently the authors of [23] provide a feasibility preserving version of Kelly’s methodology in [10]. Their algorithm introduces a slave that gives at each (master) iteration an optimal solution of a weighted proportionally fair resource allocation problem that is explicitly addressed in only the two cases of polymatroidal and flow aggregating networks. In fact, our paper contributes to this problem by proposing an efficient real-time version of the slave process, for any topol-ogy, preserving feasibility at each (slave) iteration. Amongst approximative approaches, one can quote the very recent work [14] where a multiplicative approximation for α 6= 1 and additive approximation for α = 1 is provably obtained in poly-logarithmic time in the problem parameters. Moreover, starting from any point, the algorithm reaches feasibility within poly-logarithmic time and remains feasible forever after. The algorithm described in our paper solves the problem optimally and reaches feasibility as from the first iteration from any starting point.
The work around ADMM is currently flourishing. The O(n1) best known convergence rate of ADMM [8] failed to explain its empirical fast convergence until very recently, for instance in [6], where global linear convergence rates are established in four scenarios of the strongly convex case. ADMM is also well-known for its performance that highly depends on the parameter tuning, namely, the penalty parameter ρ (or reciprocal penalty parameter λ = 1/ρ) in the augmented Lagrangian formulation (see Section III-B below). An effective use of this class of algorithms cannot be decoupled from an accurate parameter tuning, as convergence can be extremely slow otherwise. Thus, in the same paper [6], the authors provide a linear convergence proof that yields a convergence rate in a closed form that can be optimized with respect to the problem parameters. Therefore, thanks to these works, we derive in our paper a near-optimal tuning of ADMM for the α-fair resource allocation problem. Several papers use the distributivity of ADMM to design efficient distributed algorithms solving consensus formulations for e.g. model predictive control [17] and resource allocation in wireless
virtual networks [13] but do not address this fundamental detail.
To the best of our knowledge, we are the first to show how ADMM can help designing real-time distributed algorithms for computing α-fair resource allocations in distributed settings. We are also the first to exhibit a near-optimal convergence rate of ADMM in this situation with our reciprocal penalty parameter adaptation scheme.
III. FAIR RESOURCE ALLOCATION PROBLEM
In this section, we reformulate the α-fair resource allocation problem as a convex optimization problem. Then, we start off with our algorithm design by presenting C-ADMM, an algorithm that solves our problem in a centralized fashion and that will be helpful to design our distributed algorithm. A. Problem reformulation
Let R be a set of connection requests over a network with a set J of capacitated links. Each link j ∈ J has a total capacity of Cj ∈ R+. Each request r is represented by a
route containing a subset of J that, without any confusion, we still denote as r. With some abuse of notation, we write j ∈ r or r ∈ j to say that link j belongs to the route r, or route r goes through link j, respectively. Given the set of requests and their corresponding utility function fr, the network allocates
bandwidth to all the requests in order to maximize the overall utility f = L
rfr, while satisfying feasibility, i.e., the link
capacity constraints. Denote by xr the capacity allocated to
route r, and let x = (xr)r∈R. Then, we have the classic
capacity constraint in matricial form:
Ax ≤ C (1) where A = (ajr)jr is the link-route incidence binary matrix:
ajr=
1 if j ∈ r 0 otherwise.
Our aim is to compute an α-fair capacity allocation x: max
x≥0,Ax≤Cf
α(x) (2)
where the α-fair utility function fα is defined according to the Mo and Walrand’s classic characterization in [16], that we report below.
Definition 1 ((w, α)-fairness, [16]). Let F ⊂ Rn
+ be a
non-empty feasible set not reduced to {0}. Let w ∈ Rn+and x∗∈
F . We say that x∗is (w, α)-fair (or simply α-fair when there is no confusion on w) if the following holds:
∀r ∈ [1, n], x∗r> 0 and ∀x ∈ F, n X r=1 wr xr− x∗r x∗α r ≤ 0. Equivalently, x∗ is (w, α)-fair if, and only if x∗ maximizes the α-fair utility function fαdefined over F − {0}:
fα(x) = n X r=1 frα(xr), where frα(xr) = ( wr x1−α r 1−α, α 6= 1, wrlog(xr), α = 1.
The success of α-fairness is due to its generality: in fact, for α = 0, 1, 2, ∞ it is equivalent to max-throughput, proportional fairness, min-delay, and max-min fairness, respectively. We observe that the α-fair utility functions are non-decreasing, strictly concave, non-identically equal to −∞, and upper semi-continuous. It is well-known that under these conditions, the function fα admits a unique maximizer over any convex closed non-empty set.
From now on, we adopt the convex optimization termi-nology. Define for each r ∈ R the convex cost function gr: xr 7→ gr(xr) := −fr(xr). Then, g :=Lrgr= −fα
is a convex closed proper2 function over R|R|+ . Let us
in-troduce ι as the indicator function of the convex closed set {Ax ≤ C, x ≥ 0}x:
ι(x) =
0 if Ax ≤ C ∞, otherwise.
Then our α-fair problem can equivalently be formulated as the following convex program:
min x,z X r∈R gr(xr) + ι(z), (3) s.t. x − z = 0. (4) B. ADMM as an augmented Lagrangian splitting
Let us begin by recalling to the reader the basic principles of the Alternating Direction Method of Multipliers (ADMM), applied to our α-fair problem. To this aim, the augmented Lagrangian with penalty λ−1 > 0 for problem (3-4) writes3
Lλ−1(x, z, u) = g(x) + ι(z) + uT(x − z) +
1
2λ||x − z||
2
(5) where u is the vector of Lagrange multipliers. The method of multipliers consists in the following update rules, where the superscript k denotes an iteration count:
(xk+1, zk+1) = arg min x,z Lλ−1(x, z, uk) (M1) uk+1= uk+1 λ(x k+1 − zk+1). (M2) The main idea in alternating directions is in fact to decouple the variables (x, z) in the optimization stage M1: instead of a global optimization over (x, z), we only optimize Lλ−1
with respect to the variable x, then, given the new update of x, we optimize Lλ−1 with respect to z. Before stating the
corresponding update rules of ADMM, let us first remind the following Fact.
Fact 1 ([3]). Let h : Rn → ¯R = R ∪ {∞} be a closed proper convex function. The set dom(h) denotes the domain of h, that is the set upon which h takes real values. Assume dom(h) 6= ∅. Then, the following facts hold:
(i) For u ∈ Rn,λ ∈ R∗+, the minimization problem
u∗λ= arg min x h(x) + 1 2λ||u − x|| 2
2closedstands for lower semi-continuous and proper means non-identically
equal to ∞
admits a unique solution. The (λ-scaled) proximal oper-ator of h is the well-defined map proxλh: u → u∗λ. (ii) Assume that h takes the form h(x, y) = h1(x) +
h2(y), for (x, y) ∈ Rp× Rn−p (write h = h1L h2)
where h1, h2 are both closed, proper and convex.
Then, for (u, v) ∈ Rp × Rn−p, prox
λh(u, v) =
proxλh1(u), proxλh2(v).
(iii) Assume thath is the indicator function of a closed convex non-empty set F . Then PF := proxλh is the Euclidean
projection ontoF .
The definition of a proximal operator being set, a straight-forward calculus shows that we have:
∀x, u arg min z Lλ−1(x, z, u) = proxλι(x + λu) ∀z, u arg min x Lλ−1(x, z, u) = proxλg(x − λu).
ADMM can thus be expressed in the proximal (λ-scaled) form, which we refer to as Centralized ADMM (C-ADMM). Algorithm 1 Centralized ADMM (C-ADMM)
Input: Initial values z, v
1: while a suitable termination condition is not met do
2: x ← proxλg(z − v)
3: z ← P (x + v)
4: v ← v + x − z
5: end while
In Algorithm 1, P = proxλι is the projection on {Ax ≤ C, x ≥ 0}x, and v = λu the λ-scaled dual variable. Now, the
first step of Algorithm 1 (line 2) can be separated thanks to the separability property of the objective function, see Fact 1. In fact, g is fully separable, as g(x) = P gr(xr). Thus, the
proximal update of line 2 takes the trivially parallelized form: ∀r xk+1r = proxλgr(z
k
r− ukr) (6)
such that each local variable xrcan be computed separately.
Through expression (6), we are thus able to provide an efficient update rule for x, provided that the separate proximal computations are inexpensive. However, two main issues arise. Main issues with C-ADMM: a) First, an update of the variable z in line 3 of Algorithm 1 requires full knowledge of the projection mapping, which in turn requires full information on the capacity set of the network. Thus, this global update rule represents an important limiting factor to the design of a fully distributed algorithm, which is our main design interest here to follow the distribution of SDN control planes.
b) Moreover, although the convergence of C-ADMM may only require some tens of iterations (see Section VI for further details), it may be slow in terms of computation time due the successive application of a projection algorithm that would not scale with respect to the problem size. This also gives rise to a double loop algorithm where each iteration requires the convergence of an inner process that can be time-consuming. Indeed, computing the projection of a generic point onto a closed convex non-empty polyhedron is in general non-trivial. Hence, for general polyhedra, one has to operate alternate
projections, summon quadratic programming solvers or use iterative algorithms such as the one in [9].
We address issues a,b) in the next section, where we propose FD-ADMM, a distributed version of C-ADMM.
IV. THE GENERAL CONSENSUS FORM OFADMM:AN EFFICIENT DISTRIBUTED ALGORITHM DESIGN
In this section, we show how to alleviate the cost of the global projection sub-routine in C-ADMM (line 3) by decomposing the formulation with respect to the network links of each SDN domain in the fashion of a consensus problem, and present FD-ADMM. As stated at end of Section III, the global knowledge of the topology and the computational effort required by the projection step (line 3) of C-ADMM are not affordable in the distributed SDN control plane. Thus, the decomposition permits to respect the locality of the different domain controllers that now handle the projections link by link efficiently and in parallel. The decomposition into domains can be orchestrated by the SDN architect without any constraint. Unavoidably though, domains will need to exchange informa-tion as routes may traverse different domains.
A. Preliminaries
We organize the network into several domains Jp, p =
1 . . . P such that (Jp)p forms a partition of the set of links
J . Let Rp be the set of routes traversing the domain Jp via
some link j ∈ Jp. More formally, Rp = {r ∈ R : ∃ j ∈
Jp s.t. j ∈ r}. Hence, (Rp)p forms a covering of R. Let ιj
denote the indicator function for link j ∈ Jp, i.e.,
ιj(x) =
0 if P
r∈jxr≤ Cj
∞ otherwise. (7) Also, let us define Sj:= dom(ιj). Thus, for each j ∈ J , Sj
is the (convex, closed) capacity set of the link j. Finally, for j ∈ J and z ∈ RR, PROJECTION(j, z) denotes the Euclidean projection of z onto Sj.
B. Consensus form
We can now reformulate our objective to a fully separable form. For ease of notation, the variable x will be written z0and
we define R0= R. We also define an additional variable, ˜zr,
that will represent the consensus value of z0r found for each
route r over all the domains handling the route r. We write Ir= {q ∈ [0, P ] r ∈ Rq} to design the set of domain indices
(including index 0) which r belongs to. In the same fashion as in Section III-B, we plug the feasibility constraints into the objective. Each constraint being now handled separately, we can formulate Problem (3), (4) as follows:
minX r∈R gr(z0r) + X j∈J ιj(z0). (8)
In order to obtain a separable objective and fully benefit from the separability property in Fact 1, we artificially create a copy of the variable z0 for each link j. This variable will
be handled by the unique domain Jp containing j. For each
j, let zj∈ R|R| be the copy of z0 for link j.
Creating a complete copy of all the variables for each domain is, nevertheless, of no use. Each domain indeed only
needs information and manipulation over the only variables associated with the routes that they handle completely (the route is included in the domain’s links) or partially (the route meets other domains). Now, ιj actually depends only
on the sub-variable (zj)Rj def
= (zjr)r∈Rj. We erase all the
information that is irrelevant to region Jp: zj∈ RRj. We can
thus write the objective as follows: min G(z) =X r∈R gr(z0r) + X j∈J ιj((zj)Rj). (9)
To sum up, we have artificially separated the objective function by creating a minimal number of copies of the primal variable z0 in order to fully distribute the problem. Now,
instead of a global resource allocation variable, several copies of the variable account for how its value is perceived by each link of each domain. To enforce an intra- (local) and inter-(global) domain consistent value of the appropriate allocation, consensus constraints are added to the problem. This new formulation can be interpreted as a multi-agent consensus problem formulation where route r has cost gr, and link j
has cost ιj. As we separated the global objective on purpose,
the separability property of the proximal operator thus gives the following:
proxλG(u) = (proxλgr(u0r))r, (PROJECTION(j, uj))j .
These considerations permit next to write our final distributed consensus model where each agent only has access to local information.
C. Fast Distributed ADMM
We can finally distribute ADMM by putting into practice the tricks described in the previous section. Then, the general consensus form of the problem can be expressed as follows.
minX r∈R gr(z0r) + X j∈J ιj(zj) (10) zjr= zlr ∀r ∈ Rj∩ Rl ∀j, l ∈ {0} ∪ J (11) where zj = (zjr)r∈Rj ∈ R |Rj|
+ . By applying ADMM to this
formulation and using again Fact 1 we obtain, after some simplification, Algorithm 2 (Fast Distributed (FD)-ADMM). To update the consensus variables ˜zr, we exploit the fact that
the Euclidean projection of a point y ∈ Rnonto the diagonal is simply its average n1P yi1. Hence, if I denotes the indicator
function of the feasible set (11), we have: ∀r ∈ R proxλI(u)r=
1 |Jr| + 1 X l∈r ulr+ u0r ! . This yields the simple update rules at lines 4 and 104. Notably, even in the distributed case, each domain p can compute at each iteration a globally feasible allocation z∗r
for each of the routes r ∈ Rp (see Proposition 1).
Communication among domain controllers: In FD-ADMM, only domains that do share a route together have to com-municate. The communication procedures among the domain
4These updates rules are also simplified using the straightforward fact that
the sumP
l∈rulris constant. It can thus be fixed to 0 by initialization.
Algorithm 2 Fast Distributed ADMM (FD-ADMM)
1: procedureOF DOMAINp
Input: Reciprocal penalty parameter λ, (gr)r∈Rp
2: RECEIVE zqr, z∗qr ∀q ∈ Ir ∀r ∈ Rp
3: ENFORCE z∗r= minq∈Irz∗qr ∀r ∈ Rp
4: z˜r← |J1 r|+1 P q∈Irzqr+ z0r ∀r ∈ Rp 5: for j ∈ Jp∪ {0} do 6: ujr ← ujr+ zjr− ˜zr ∀r ∈ Rj 7: zj← PROJECTION(j, ˜z − uj) 8: end for 9: z0r← proxλgr(˜zr− u0r) ∀r ∈ Rp
10: SEND zpr = Pj∈Jr∩Jpzjr and z∗pr = minj∈Jpzjr
to domains q ∈ Ir ∀r ∈ Rp
11: end procedure
controllers are described at lines 2 and 10. In these steps, the domains gather from and broadcast to adjacent domains the sole information related to routes that they share in common. In particular, domains are blind to routes that do not traverse them, and can keep their internal routes secret from others. In details, after each iteration of the algorithm, each domain Jpreceives the minimal information from other domains such
that Jp is still able to compute a local value zpr and a locally
feasible value z∗pr. Next, Jp send them back to neighboring
domains Ir that r traverses.
Communication overhead: In terms of overhead, we can easily evaluate the number of floats transmitted between each domain at each iteration. At each communication, domain Jp must transmit zpr and z∗pr for each r ∈ Rp to each
other domain that r traverses. The variable z0 does not need
to be centralized or transmitted between controllers. Each domain controller may actually have a copy z0 and perform
the (low-cost) computation of their update rule (see line 9 in Algorithm 2) locally. Hence, domain p transmits in total 2P
q6=p|Rp∩Rq| floats to the set of its peers. As a comparison,
in a distributed implementation of the algorithm given in [15] and stated in Section VI, each domain p transmits in total P
q6=p|{j ∈ Jp, ∃r ∈ Rq s.t. j ∈ r}| floats to the set of its
peers, which is bounded by (P − 1)|Jp| as |R| grows.
Feasibility preservation: A potential drawback of the dis-tributed approach is the potential feasibility violation by the iterate ˜zk. However, we have the following positive result. Proposition 1. FD-ADMM provides a sequence of feasible points that converges to the optimum.
Proof. Consider the iteration number k and drop the super-script k for lightness. For any link j, we have by line 7 of Al-gorithm 2 that zjis feasible in link j. That is,Pr∈jzjr ≤ Cj.
Define z∗r= minj∈Jrzjr. Then, for each link j:
X r∈j z∗r≤ X r∈j zjr ≤ Cj. (12)
Thus, no capacity is violated by the allocation z∗r. At the
optimum, the consensus is reached. Thus (z∗rk )k is a feasible
The number z∗r introduced in Proposition 1 above in fact
corresponds to the introduced variable of the same name FD-ADMM. Thus, in a certain way, for sufficiently loaded and communicating domains (i.e. the |Rp ∩ Rq| are large
enough) we sacrifice some overhead (counted on a per iteration basis) compared to standard dual methods, but in exchange for anytime feasibility, a major feature that dual methods do not generically provide.
V. IMPLEMENTATION AND ALGORITHM TUNING
In this section, we discuss two major points in the design of FD-ADMM. First, we precise and justify the choice of the procedure PROJECTION, in line 7 of FD-ADMM Algorithm 2. Next, we derive an explicit adaptive update of the reciprocal penalty parameter λ that permits to accelerate the convergence of FD-ADMM on any instance.
A. Projection procedure: A discussion
In Section IV, we advocated a link-wise separation of the formulation because it is non-trivial to project an arbitrary point onto an arbitrary closed convex polyhedron. However, the projection onto the sets Sj (see Section IV-A) can be
done with an exact method with a complexity dominated by the one of sorting a list of the size of its dimension. In average, sorting a list of length q is done in O(q log q). Hence, by operating instead a link-by-link projection, the controllers save a huge amount of time by providing an (generically infeasible) approximate projection point zpr and deriving a
locally feasible allocation z∗pr (see Algorithm 2 line 10).
Although the quality of the global iterate z∗ may be altered
by further distribution of the projection, the point is quickly generated. Paradoxically enough, FD-ADMM therefore fully adapts to any network distribution into domains because it functions by link, regardless of the network partition into domains. The algorithm we use for PROJECTION in FD-ADMM is presented for instance in [5] in which the authors also give a correctness proof and performance demonstration. It permits to provide an efficient update for each domain Jp.
B. Estimating the optimal parameterλ
It is well-known that the reciprocal penalty parameter λ highly conditions the convergence speed of ADMM. An inaccurate tuning can indeed lead to a very slow convergence. For appropriate problems, it is possible to use a result proven in [6] to compute an optimal reciprocal penalty parameter, that we here report. It will help us tune FD-ADMM to optimize its convergence performance5.
Theorem 1 ([6]). Assume that the following problem: min F (x) + G(z) (M) s.t. M x − P z = 0
has a saddle point, and both objective functions are convex. Assume that M has full row rank, and that F is σ-strongly convex and has aL-Lipschitz gradient. Then, the sequence of
5We recall that a differentiable function f : Rn→ ¯R is strongly convex
with modulus σ if (∇f (x) − ∇f (y))T(x − y) ≥ σ||x − y||2, ∀x, y ∈
dom(f ). Moreover, f is Lipschitz with modulus L if |f (x) − f (y)| ≤ L|x − y|, ∀x, y ∈ dom(f ).
iterates (primal and dual concatenated) of ADMM converges linearly with rate(1 + δ)−1, where6
δ = 2 ||M || 2 λσ + Lλ λmin(MTM ) −1
and λ1 is the penalty parameter in the augmented Lagrangian form (see Section III-B).
The following result directly follows.
Corollary 1. The optimal reciprocal penalty parameter is λ∗=
r ||M ||2λ
min(MTM )
σL .
In order to be able to apply Corollary 1, we still need to express the coefficients of interest σ, Ld. Due to page
limitation, we defer their calculus to [1] and we here report the result.
Fact 2. The function g =L
rgrisσ-strongly convex and has
Ld-Lipschitz gradient with:
σ = α min r wr Brα+1 , Ld= α max r wr dα+1r
on any compact subset of dom(g) of the form Kd = {x ≥
d, Ax ≤ C}, for d 0, and Br= minj∈rCj.
Unfortunately, Corollary 1 cannot be directly applied to our general consensus formulation. Indeed, its matricial formula-tion does not provide a full-row rank matrix M . The problem which the Theorem 1 applies to is actually the original, centralized one in (3-4). Therefore, we will derive a reciprocal penalty parameter selection for the centralized problem, and use it as a tool to estimate a satisfactory parameter for FD-ADMM.
However, the last difficulty we encounter in choosing the optimal reciprocal penalty parameter is to correctly evaluate the Lipschitz modulus. Unfortunately, ∇g is not Lipschitz on the feasibility set, because of the singularity of each gr at 0.
In order to circumvent this problem, we introduce the classic concept of disagreement point d, according to bargaining theory terminology. A disagreement point d represents the minimal values for an allocation of each route. This allows to reduce the feasibility set to a compact subset of the form Kd, d 0, on which ∇g is now Lipschitz. The disagreement
point can be naturally defined as the feasible point z∗ at the
first iteration. Generically, there is no a priori guarantee that the set Kz∗ contains the optimum, but, we remark that at
least in the first iterations, the use of z∗ provides a good
approximation of the best reciprocal penalty parameter. The analytical evaluation of this phenomenon goes beyond the scope of this paper and we keep it for future work.
Thus, finally, we update λ in an adaptive fashion in the beginning of the algorithm with the help of those points. We found empirically that operating such update only at the initial steps of FD-ADMM and then fixing λ for the rest of the execution provides a good performance in terms of convergence speed. In the next section, we describe this
6λ
min is the smallest eigenvalue of a positive matrix, and ||M || is the
Fig. 1: C-ADMM against FD-ADMM: execution time and iteration count.
typical phenomenon in Figure 2. In all our simulations, we use the simple following update scheme to estimate the optimal penalty parameter at each execution of the algorithm. Scheme 1 (Reciprocal Penalty Adaptation). Set threshold τ . At all iterations below τ , denote by p the last output of a feasible point. Then, choose the new reciprocal penalty parameter as: λ∗ = 1 α min r∈R wr Brα+1 max r∈R wr pα+1r −12 . After τ iterations, do not update λ.
In our numerical evaluations we will set τ = 30. Thus, FD-ADMM is now fully tuned and we are ready to demonstrate its performance in the next section, in terms of convergence speed in real-time scenarios.
VI. PERFORMANCE ANALYSIS
We now evaluate numerically FD-ADMM in terms of its convergence properties. More specifically, in Section VI-A we compare the performance of FD-ADMM and C-ADMM in offline scenarios where the optimum is desired. In Section VI-B we evaluate FD-ADMM in real-time scenarios, where good and feasible solutions are needed on-the-fly as weights wr vary over time. In order to benchmark the transient
properties of FD-ADMM we use the standard Lagrangian dual decomposition approach (LAGR) for single-path routing in [25, 15, 19], that we recall in Algorithm 3. We here assume that domain controllers operate in synchronous mode. In this case, the decomposition into domains has no impact on FD-ADMM performance, as projection is on a link-basis. All simulations are made for the proportional fairness objective functions (α = 1). We used the proximal operation formu-las found in [3]. The algorithms under investigation were evaluated using BT’s 21 CN network topology7, containing
106 nodes and 474 links. The requests were generated by computing the shortest path between randomly chosen sources and destinations.
7We would like to thank the authors of [15] for their willingness to share
the BT 21 CN topology dataset.
Fig. 2: Convergence rate of FD-ADMM vs. reciprocal penalty param-eter. The adaptive approximation demonstrates sufficient accuracy.
A. Algorithm design
Evaluating the alleviation of the compute-intensive parts of C-ADMM was a key concern to motivate and validate the distribution to FD-ADMM. To this aim, we show in Figure 1 the computation time and iteration count for those two algorithms on small instances for a number of requests ranging from 1 to 200. The centralized projection in C-ADMM is executed using the variation of Hildreth’s projection algorithm on general polyhedra in [9]. When convergence is desired, a precise stopping criterion for FD-ADMM is available, as the optimality gap can be upper-bounded by the primal and dual residuals, see [3]. In our case, evaluating those residuals results in computing the absolute variation of two consecutive values
Algorithm 3 Lagrangian-based gradient descent (LAGR) Input: Initial positive values uj
while a suitable termination condition is not met do xr← arg max x≥0 {fr(x) − xPj:j∈ruj)} ∀r uj = uj−2Cuj j(Cj− P r:j∈rxr) ∀j end while
Fig. 3: Iteration count for FD-ADMM vs the mean link load (average value of the |Rj|).
of ˜z, and the consensus accuracy8 max
r∈R,j∈r|zjr − ˜zr|.
This is a first advantage for FD-ADMM implementation as no robust stopping criterion is available for standard gradient descent. When an optimality gap is computed, we thus con-sider a 10−6-approximation by FD-ADMM as the reference for all tested algorithms. In Figure 2, we illustrated, on a small instance with 200 requests, the number of iterations of FD-ADMM to reach convergence for a various number of the parameter values, in order to evaluate our adaptive scheme’s accuracy with respect to the empirically best found parameter. It shows that our approximation of λ∗ is fairly satisfactory.
In Figure 1, FD-ADMM shows that distributing the consensus over the links exchanges several more iterations for a reduction of the compute time by two orders of magnitude for small instances. Hence, the distribution does not seem to cost too much convergence rate. Not surprisingly, the use of a central projection sub-routine makes C-ADMM impossible to scale. The convergence criterion used in Figures 1 and 3 is modest (10−1). Finally, we plotted a notable behavior of FD-ADMM in Figure 3. One can imagine a link between the convergence rate and the mean link load, i.e., [J |1 P |Rj|. This conjecture
requires further investigation that we keep for future work. B. Comparison against Lagrangian method
We now compare the proposed FD-ADMM algorithm against the classic LAGR Algorithm 3, see [25, 15, 19]. To this aim we perform two experiments, in real-time and static scenarios, respectively.
We start by evaluating the real-time responsiveness of FD-ADMM by considering a small scenario where 200 routes are established and the weights (wt
r)r∈R,t∈0...T vary over discrete
time t, following the formula:
wrt+1∈ [(1 − a)wtr, (1 + a)w t
r] a ∈ [0, 1],
where at each event t, wrt is chosen uniformly within the
above interval in which a determines the amplitude of the weight variation. In Figure 4 we illustrated the average op-timality gap of the two algorithms achieved over 20 events
8One can choose any other norm inL R|Rj|
.
Fig. 4: Average optimality gap E[gap] vs. the variation amplitude a.
with 10 iterations between each event. We observe that FD-ADMM outperforms LAGR in terms of optimality gap, al-though the performance of both algorithms is fairly acceptable. However, remarkably, FD-ADMM remains always feasible whereas LAGR constantly violates the constraints as weights wr change in real-time. Figure 5 shows the percentage of
constraints of the problem that are violated for each value of the amplitude a. In fact, LAGR iteratively approaches the fair resource allocation from the outside of the feasible set. This drawback is commonly amended by projecting the solution onto the feasible set. However, this is not doable in our distributed setting, as projection requires costly on-the-fly operations that require full topological information. For such reasons, we claim that the standard LAGR algorithm is not well suited for computing real-time fair allocations in a distributed SDN setting.
In our last experiment we test the two algorithms under a static scenario, where the weights wrdo not vary over time and
LAGR has enough time to find at least one feasible solution. In Figure 6 we compare the optimality gap of the best feasible solutions found after 5 seconds runtime by FD-ADMM and LAGR, for different instance sizes over BT topology. We observe that FD-ADMM obtains a close-to-optimal feasible solution for all the instance sizes (from 100 to 6000 requests),
Fig. 5: Average percentage of violated constraints E[v] by LAGR vs. the variation amplitude a.
Fig. 6: Optimality gap of the best feasible point found after 5 seconds runtime.
while LAGR is still far from the optimum especially when the instance becomes large.
To recap, in this section we have demonstrated by exper-imentation that FD-ADMM reacts quickly to unpredictable network variations, while preserving the feasibility of the solutions computed iteratively. We then claim that FD-ADMM is a good candidate for real-time fair resource allocation in distributed SDN scenarios.
VII. CONCLUSIONS AND FUTURE WORK
In this paper we addressed the real-time fair resource allocation problem in the context of a distributed SDN control plane architecture. Our main contribution is the design of a distributed algorithm that continuously generates a sequence of feasible solutions and adapts to any partitioning of the network into domains. We reformulated the α-fair resource allocation problem in the fashion of a general consensus prob-lem to derive the FD-ADMM algorithm. This algorithm can be massively parallelized on several processors that manage different regions of the network, hence fully benefiting from the computing resources of SDN controllers in distributed architectures. We also provided a strategy for a near-optimal estimation of the penalty parameter of FD-ADMM that boosts its convergence. Finally, we compared FD-ADMM to a stan-dard dual Lagrangian decomposition method (LAGR) and we demonstrated how the former is more adapted to a real-time situation where bandwidth has to be adjusted on-the-fly. In fact, FD-ADMM ensures a smaller optimality gap since the very first iterations and, most importantly, it produces a feasible fair allocation at all iterations.
As a next step, we envision to adapt our formulation to the case where multiple candidate paths are available for each request. Moreover, we plan to run FD-ADMM asynchronously while still guarantying near-optimal convergence rate and anytime feasibility.
ACKNOWLEDGMENT
This is the author edited copy of the article published in the Proceedings of ITC 29, Genoa, 4-8 September 2017.
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